12 Linear Programming

 

IGDEMX ZXBAXMLLGDA

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IGDEMX ZXBAXMLLGDA >.=

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HM^GN NBDNEZP^ =.

Igdemr gdequmtgbds tfe gdequmigtges bc tfe cbrl mx + hy £ n, mx + hy 9 n, mx + hy ¾ n md` mx + hy 4 n mre nmiie` igdemr gdequmtgbds.

8.

Igdemr prbarmllgda Gt gs tfe letfb` use` gd `engsgbd lmogda lmog da gd husgdess cbr bhtmgdgda tfe iemst nbst md` lmxglul prbcgts tfrbuaf igdemr expressgbds exp ressgbds suhkent tb nertmgd agved agv ed nbd`gtgbds nmiie` igdemr igd emr nbdstrmgdts.

BHKENPGSE CUDNPGBD ;  M igdemr ccudntgbd udntgbd bc tfe gdvbive` vmrgmhies wfgnf we wmdt tb lgdglgze br lmxglgze suhkent tb tfe agved igdemr nbdstrmgdts gs odbwd ms bhkentgve cudntgbd. =.

Bptglmi vmiue Pfe lmxglul (lgdglul) vmiue bc md bhkentgve cudntgbd gs odbwd ms gts bptglmi vmiue.

8.

Cemsghie sbiutgbd M set bc vmiues bc tfe vmrgmhies smtgscygda smtgscyg da mii nbdstrmgdts gs o odbwd dbwd ms m cemsghie sbiutgbd.

>.

Bptglmi sbiutgbd M cemsghie sbiutgbd wfgnf iem`s tb tfe bptglmi vmiue (lmxglul br lgdglul) lgdglu l) bc md bhkentgve cudntgbd gs nmiie` gts bptglmi sbiutgbd.

0.

Bptglgzmtg bd tenfdgques Pfe prbness bc bhtmgdgda tfe bptglmi vmiues mre nmiie` bptglgzmtgbd bptglgzmtgb d tenfdgques. tenfdgques.

5.

Cemsghie reag bd Pfe nbllbd reagbd `eterlgde` hy mii tfe nbdstrmgdts md` dbddeamtgvgty deamtg vgty restrgntgbd restrg ntgbd bc tfe prbhiel prbhiel gs nmiie` cem cemsghie sghie reagbd. reagbd.

3.

Igdemr Zrbarmllgda.Zrbhiel(IZZ) M I.Z.Z I.Z.Z.. nbdsgsts bc lmxglg lmxglgzgda zgda br lgdglgzgda md bh kentgve cudntgbd suhkent tb nertmgd nbdstrmgdts.

1.

Gc tfe cemsghie reagbd gs udhbud`e` tfed m lmxglul br lgdglul lmy dbt exgst. Fbwever gc gt exgsts, gt g t lust bnnur mt nbrder pbgdt (vertex) bc X.

CUD CUD@MLED MLEDPM PMI I PFEB FEBXEL^ EL^ ; Pfebrel =.  =. Iet X he tfe cemsghie reagbd (nbdvex pbiyabd) cbr m igdemr prbarmllg prba rmllgda da prbhiel md` iet z < mx + hy he tfe bhkentgv bhkentgve e cudntgbd. Qfed z fms md bptglmi vmiue (lmxglul br lgdglul) wfere tfe vmrgmhies x md` y mre suhkent tb nbdstrmgdts `esnrghe` hy igdemr i gdemr gdequmigtges, tfgs bptglmi vmiue lust bnnur mt m vertex (nbrder pbgdt) bc  tfe cemsghie reagbd. cemsghie reagbd reag bd cbr m igdemr prbarmllgda prbhiel md` iet z < mx + hy Pfebrel 8. Iet 8. Iet X he tfe cemsghie he tfe bhkentgve cudntgbd. Gc X gs hbud`e`, tfed tfe bhkentgve cudntgbd z fms hbtf lmxglul lmxg lul md` lgdgul vmiue bc X md` emnf bc tfese bnnurs mt m vertex (nbrder pbgdt) bc X.

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NBXD NBXDEX EX ZBG BGDP DP LEPFB EPFB@ @ ; Pfgs letfb` nblprgses bc tfe cbiibwgda steps ; (g) G`edtgcy tfe udodbwds (`engsgbd vmrgmhies) vmrgmhi es) md` `edbte tfel hy x md` y. (gg) Cbrluimte tfe bhkentgve cudntgbd gd g d terls bc x md` y (ggg) Prmdscbrl rmdscbrl mii nbdstrmgdts nbdstrmgdts tb iigdemr gdemr gdequmtgbds (vg) ^bive tfese gdequmtgbds md` cgd` tfe cemsghie cemsghi e reagbd. (v) @eterlgde tfe nbrder pbgdts bc tfe cemsghie reagbd. Iet L md` l he tfe imraest md` slmiiest vmiues mt tfese pbgdts (vg) Gc tfe cemsghie reagbd gs hbud`e`, L md` l mre respentgveiy lmxglul md` lgdglul vmiues bc tfe bhkentgve cudntgbd. Gc tfe cemsghie reagbd gs udhbud`e`, tfed (g) L gs tfe lmxglul vmiue bc tfe bhkentgve cudntgbd, gc tfe bped fmic pimde `eterlgde` hy mx + h hy y 4 L fms db pbgdt gd nbllbd wgtf tfe cemsghie reagbd. Btferwgse tfe bhkentgve cudntgbd fms db lmxglul vmiue. (gg) l gs tfe lgdglul vmiue bc tfe bhke bhkentgve ntgve cudntgbd, gc tfe bped fmic pimde `ete `eterlgde` rlgde` hy mx + hy 9 l fms db pbgdt gd nbllbd wgtf tfe cemsghie reagbd. Btferwg Btferwgse se tfe bhkentgve cudntgbd fms db lgdgul vmiue.

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IGDEMX ZXBAXMLLGDA

8

^BISE@ ZXBHIE^ Ex.=

Md bgi bgi nblp nblpmd mdy y fms twb twb `e `epb pbts ts M md md` ` H wg wgtf tf nmpmn nmpmngt gtges ges bc 1: 1::: :: I md` md` 0: 0::: :: I respentgveiy.. Pfe nblpmdy gs tb suppiy bgi tb tfree petrbi pulps, @, E md` C wfbse respentgveiy requgreledts requgr eledts mre 05::I, >:::I md` >5::I md` >5::I respe respentgve ntgveiy. iy. Pfe `gstmdne `gstmdnes s (gd ol) hetweed tfe `epbts md` tfe petrbi pulps pu lps gs agved gd tfe cbiibwgda tmhie ; @gstmdne (gd ol) C rb l / P b @

M 1

H >

E

3

0

C

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8 

 Mssulgda  Mssul gda tfmt tfe trmdspb trmdspbrtmtg rtmtgbd bd nbst bc =: igtres igtres bc bgi gs Xs = per ol, fbw sfb sfbui` ui` tfe `eigvery he snfe`uie` snfe `uie` gd br`er tfmt tfe trmdspbrtmtgbd nbst gs lgdglul 2 Qfmt gs tfe lgdglul nbst 2  ^bi.

Iet x igtres md` y igtres i gtres bc bgi he supp suppige` ige` crbl `epbt M tb petrbi p pulps ulps @ md` E respentgveiy. Pfed (1,::: „ x „ y) igtres bc bgi gs suppige` supp ige` tb petrbi pulp C crbl `epbt M. Fedne, x ¾ :, y ¾ : md` x + y £ 1::: ^gdne tfe requgreledt bc petrbi pulps @, E md` C mre

@epbt M

05:: igtres, >::: igtres md` >5:: igtres respentgveiy,

x

tfe `epbt H wgii suppiy (05:: „ x) igtres, (>::: „ y)

Zetrbi Zulp @

md` (>5:: „ 1::: + x + y) igtres igtr es tb tfe petrbi pulps @, E md` C respentgveiy.

y Zetrbi Zulp E

05::„x

>:::„y

@epbt H

Misb, (05:: „ x) + (>::: „ y) + (>5:: „ 1::: + x + y) y)

05:: „ x „ y Zetrbi Zulp C   :  :    1     „     y     :  :   x  +   >  5   +

< 0:::, wfgnf gs equmi tb tfe t fe nmpmngty bc tfe `epbt H. Fedne,, 05:: „ x ¾ :, >::: „ y ¾ : md` x + y „ >5:: ¾ : Fedne x £ 05::, y £ >::: md` x + y ¾ >5::.

g.e.,

Pfus, tfe IZZ gs Lgdglgse ; N <

 +

>(1::  - x - y ) 3y 1x  +  + =: =: =:

8(x +   y - >5:: ) 0 (>:::   - y) >( 05::   - x)  +  + =: =: =:

 < >x + y + >7,5:: suhkent tb tfe nbdstrmgdst ; x + y £ 1,:::  

x £ 0,5::

 

y £ >,:::

x + y ¾ >,5:: md`

W

1::: 3::: 5::: 0::: >:::

E

@ B

8::: =:::

 M

H

B   :  :  :   :  :  :   :  :  :   :  :  :   :  :  :   :  :  :   :  :  :   =   8   >   0   5   3    1

x ¾ :, y ¾ :

P fe fe s f fm m `e `e ` r ea ea g b bd d g d A rm rm pf pf = 8 8.. 8 86 6 g s t fe fe cemsghie reagbd wgtf nbrder pbgdts M (>5::, (>5::, :), H (05: ( 05::, :, :), : ), N (0 (05:: 5::,, 85::), @ ((0:::, 0:::, >:::) md` E (5::, >:::)

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IGDEMX ZXBAXMLLGDA

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Pfe vmiue bc N mt tfese cgve pbgdts mre ;  

N mt M (>5::, :) < 5:,:::

 

N mt H (05::, :) < 5>,:::

N mt N (05::, 85::) < 55,5::  

N mt @ (0:::, >5:::) < 50,:::   N mt E (5::, (5::, >:::) < 00,:::

Fedne, tfe lgdglul lgdglu l bc trmdspbrtmtgbd gs Xs 00:: mt (5::, >:::), sgdne tfe trmdspbrtmtgbd trmdspbrtmtg bd nbst bc =: igtres gs Xe = per ol. Pfus, 5:: igtres, >::: igtres md` >5:: igtres igtr es tb @, E md` C respentgvei respentgveiy y were suppige` suppig e` crbl `epbt M? md` 0::: igtres were suppige` suppi ge` tb @ crbl `epbt H. Ex.8 Ex. 8

M crugt crugt arbwer arbwer nmd use twb types bc certgigzer gd fgs amr`ed, hrmd` Z md` hrmd` ]. Pfe mlbudts (gd oa) bc dgtrbaed, pfbspfbrgn mng`, pbtmsf md` nfibrgde gd m hma bc emnf hrmd` mre agved gd tfe tmhie heibw. Pests gd`gnmte tfmt tfe amr`ed dee`s mt iemst 80: oa bc pfbspfbrgn pfbspfbrg n mng`, mt iemst 81: oa bc pbtmsf md` md ` mt lbst >=: >=: oa bc nfibrg nfibrgde. de.  Gc tfe arbwer wmdts tb lgdglgse tfe mlbudt b bcc dgtrbaed m``e` tb tfe tfe amr`ed, fbw  lmdy hmas h mas bc emnf hrmd` sfbui` sfbui` he use` 2 Qfmt gs tfe lgdglul mlbudt bc dgtrbaed dgtrbaed m``e` gd tfe amr`ed 2  (oa per hma) Hrmd` Z

 

 D  Dggtrbaed

>

 Z  Zf fbspfbrgn Mng`

=

 Zb  Zbtmsf

>

=.5

=.5

8

 N  Nf fibrgde

^bi.

Hrmd` ]   >.5

 

8      

Iet tfe crugt arbwer use x hmas h mas bc hrmd` Z md` y hmas bc hrmd` ]. Pfed tfe Igdemr Zrbarmllgda cbrluimte` crbl tfe agved prbhiel gs ; Lgdglgse ; D < >x + >.5y suhkent tb tfe nbdtrmgdts ; x + 8y ¾ 80: >x + =.5y ¾ 81: =.5x + 8y £ >=: md` x ¾ :, y ¾ : Pfe sfm`e` reagbd MHN gs tfe cemsghie reagbd wfgnf gs hbud`e`. hbud`e`. Pfe nbbr`gdmtes bc tfe nbrder pbgdts mre M (8:, =0:), H(0:, =::) md` N(=0:, 5:). Pfe vmiues bc D mt tfese pbgdts mre 55:, 01: md` 015.

W =6: =6: =3: =3: =0 =0: : =8: =8: =:: 6: 3: 0: 8: B

Fedne, D gs lgdglul lgdg lul mt H (0:, =::). Pfe lgdglul mlbudt bc dgtrbaed

 M(8: ,=0: ,=0:))

  H (=0:,=::)

>      x       +      =       . 5         y        Ex .>

M cmntb cmntbry ry lmoes teddgs rmnoets rmnoets md` nrgnoet hmts. M tedd teddgs gs rmnoet tmoes = =.5 .5 fburs bc  lmnfgde tgle md` > fburs bc nrmctlmd‗s tgle gd gts lmogda wfgie wfg ie m nrgnoet hmt tmoe > fburs bc lmnfgde tgle md` = fbur bc nrmctlmd‗s tgle. Gd m `my, tfe cmntbry cmn tbry fms tfe mvmgimhgigty bc dbt lbre tfmd >8 fburs bc lmnfgde lmnfgd e tgle md` 80 fburs bc nrmctlmd‗s tgle. (g) Qfmt dulher bc rmnoets md` hmts lust he lm`e gc tfe cmntbry c mntbry gs tfe wbro mt cuii  nmpmngty  (gg) Gc tfe prbcgt bd m rmnoet md` bd m hmt gs Xs 8: md` Xs =: respentgveiy, cgd` tfe

^bi.

lmxglul prbcgt bc tfe cmntbry wfed gt wbros mt cuii nmpmngty. Iet tfe cmntbry lmoe x teddgs rmnoets md` y nrgnoet hmts. Pfed, tfe IZZ nbrrespbd`gda tb tfgs prbhiel gs W Lmxglgse Z < 8:x + =:y ^uhkent tb tfe nbdstrmgdts ; 0: =.5x + >y £ 08 >8   >x + y £ 80y 80 md` x ¾ :, y ¾ : =3 @rmwgda tfe armpf bc tfe systel bc gdequmtgbds,  M(:,=  M( :,=0) 0) H(0,=8) we aet MHN ms tfe cemsghie reagbd (sfm`e` reagbd). 6 Pfe nbbr`gdmtes bc tfe nbrder pbgdts pbgd ts M, H md` N mre (:, =0), H (0, =8) md` N (6, :) respentgveiy. respentgve iy. B 6 = 3 80 >8 0: V Pfe vmiues bc Z mt M, H, N mre =0:, 8:: md` =3: N(6,:) =.5x + >y < 08 respentgveiy. >x+yy £ =8: 5: 3x + >y £ 80  80: g.e., 8x + y £ 6: N(:,0:) 0: md` x ¾ :, y ¾ : Pfe armpf bc tfgs systel bc gdequmtgbds agves >: tfe sfm`e` reagbd BMHN ms tfe cemsghie reagbd. H(>:,8:) 8: 8  x   +  >    y  :, 8:) md` (:, 0:) respentgveiy. Pfe Z mt tfese pbgdts mre respentg :, 86:,veiy. 0=: md` 0::. Fedne, Z gs lmxglul lmxg lul mt H (>:, 8:).

= : 8 : > :  M (0:.:)

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5: 3 : 1: V

8x+y: >: pmnomaes bc snrew M md` 8: pmnomaes bc  snrew H. Ex.5 Ex. 5

M lmducmn lmducmnturer turer fms tfree lm lmnfgdes nfgdes G,GG G,GG md` GGG gdst gdstmiie` miie` gd fgs ccmntbry mntbry.. Lmn Lmnfgdes fgdes G md`  md`   GG mre nmpmhie bc hegda bpermte` cbr mt lbst =8 fburs wferems lmnfgde GG lust he bpermte` cbr mt iemst 5 fburs m `my. ^fe prb`unes bdiy twb gtels L md` D emnf requgrgda tfe use bc mii tfe tfree tfre e lmnfgdes. Pfe dulher bc fburs requ requgre` gre` cbr prb`ungda = udgt  bc emnf bc L md` D bd tfe tfree lmnfgdes mre agved gd tfe cbiibwgda tmhie ;

Gtels

^bi.

Dulhers bc fburs requgre` bd lmnfgdes G

GG 

GGG 

L

=

8 

=

D

8

=

=.85 

^fe lmoes m prbcgt prbcg t bc Xs 3:: md` Xs 0:: bd gtels L md` D respen respentgvei tgveiy. y. Fbw lmdy bc  emnf tfel sfbui` sfe prb`une sb ms tfe lmxglgse fer prbcgt mssulgda mssulgda tfmt sfe nmd  sfeii mii tfe gtels tfmt sfe prb`une` 2 Qfmt wgii he tfe lmxglul lmxglul prbcgt 2  Iet x md` y he tfe dulher bc gtels L md` D respentgveiy. Pbtmi prbcgt prbcg t bd tfe prb`untgbd < Xs (3::x + 0::y) Lmtfelmtgnmi cbrluimtgbd bc tfe agved prbhiel gs ms cbiibws ; Lmxglgse Lmxgl gse [ < 3::x + 0::y suhkent tb tfe nbdstrmgdts ; x + 8y £ =  =8 8 (nbdst dstrmg mgd dt bd Lmnfgd nfgde e G) 8x + y £ =  =8 8 (nb nbds dstr trm mgd gdtt bd Lm Lmnf nfgd gde e GG) 5 (nbdst strm rmgdt gdt bd Lm Lmnf nfgd gde e GGG GGG) x + y ¾ 5 (nbd 0

.... .... ..((=) .. .... .... ..(8 (8))

W =8 (:,=8) =:

.. .... .... ..(> (>))

  x ¾ :, y ¾ : ..... .(0) Iet us `rmw tfe armpf bc nbdstrmgdts (=) tb (0). MHN@E gs tfe cemsghie reagbd ( s fm fm ` e e` `) ms sf fb b w d g d A rm rm pf pf = 8 8.. > 8 `eterlgde` hy tfe nbdstrmgdt s (=) tb (0). Bhserve tfmt tfe cemsghie reagbd gs hbud`e`, nbbr`gdmtes nbbr`g dmtes bc tfe nbrder pbgdts M, H, N, @ md` E mre (5, :), (3, :), (0, 0), (:, 3) md` (:, 0) respentgveiy. respentgv eiy. Iet us evmiumte [ < 3::x + 0::y

6 3 @(:,3) N(0,0)

E(:,0)

x  +  8    y  ,3::

(0, 0)

0::: Lmxglul

(:, 3) (:, 0)

8,0:: =,3::

Fedne, tfe lmducmnturer fms tb prb`une 0 udgts bc emnf gtel tb aet tfe lmxglul prbcgt bc  Xs 0,:::.

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IGDEMX ZXBAXMLLGDA

3

UD^BISE@ ZXBHIEL^ EVEXNG^E „ G ].=

Lmxglgze Lmxglg ze z < =:x + =5y ud`er tfe nbd`gtgbds   5x + 8y ¾ =:,   >x + 1y ¾ 8=,   1x + 5y £ >5,   y £ 0,   x, y ¾ :.

].8

M cgrl lmducmntures abi` rgdas md` nfmgds. Pfe tbtmi dulher bc rgdas md` nfmgds lmducmnture` per `my gs mt lbst >:. Gt tmoes 8 fburs tb lmoe m rg rgda da md` bde fbur cbr m nfmgd. Pfe lmxglul dulher bc fburs mvmgimhie gs 0: per `my. Gc tfe prbcgt bd m rgda gs Xs 85: md` bd m nfmgd gs Xs =5:, =5:, fbw lmdy bc bc emnf emnf sfb sfbui` ui` he lmducmnt ure` `mgi ` mgiy y sb ms tb l lmxg mxglgze lgze tfe prbcgt 2

].>

M prb`uner fms >: md` =1 udgts bc imhbur md` nmpgtmi respentgveiy, wfgnf fe nmd use tb prb`une prb`un e twb types bc abb`s abb`s,, V md` W. Pb prb`une prb`une bde u udg dg t bc b c V, V , 8 udg ud g ts bc im hbur hb ur md md` ` > ud udgt gt s bc nmpgtmi mre requgre`. ^glgimriy, > udgts bc imhbur md` bde udgt bc nmpgtmi gs requgre` tb prb`une prb`un e bde udgt bc W. W. Gc V md` W mre prgne` mt Xs =:: md` Xs =8: per udg udgtt respentgve respen tgveiy, iy, fbw sfbui` tfe prb`uner use fgs resburnes tb lmxglgze tfe tbtmi revedue. ^bive tfe prbhiel armpfgnmiiy 2

].0

M cgrl lmducmntures lmducmntures twb types bc prb`unts, M md` H, md` seiis tfel mt m prbcgt bc Xs 5 per udgt bc type M md` Xs > per udgt bc type H. Emnf prb`unt gs prb`une` bd twb lmnfgdes, L = md` L8. Bde udgt ud gt bc type t ype M requ requgre gres s bde lgdute bc prbnessgda tgle bd L = md` twb lgdutes lgdutes bc prbness prbnessgda gda tgle bd L8, wferems bde udgt bc type H requgres bde lgdute bc prbnessgda tgle bd L = md` bde lgdute bd L8. Lmnfgdes L= md` L8 mre respentgveiy mvmgimhie mvmgi mhie cbr mt lbst 5 fburs md` 3 fburs gd m `my. Cgd` but fbw lmdy udgts udg ts bc emnf type type bc prb`unt prb`unt sfb sfbui` ui` tfe cgrl prb`une m `my gd br`er b r`er tb lmxglgze tfe prbcgt. ^bive tfe prbhiel armpfgnmiiy. armpfgnmi iy.

].5 ].5

M `get `getgngmd gngmd wgsfes wgsfes fer fer nigedts nigedts tb fmve m lgdglul lgdglul bc >: udg ts bc p prb rbteg teg ds md md` ` m lmxg lm xglul lul bc =6 md` =0 udgts bc nmrhbfy`rmtes md` cmts respentgveiy. Cbb` V nbsts Xs =5 per oa md` prbvg prb vg`es `es > udgts udg ts b bcc prb teg ds, 8 u dgt s b bcc nmrhbfy`rmtes nmr hbfy`rmtes md` = ud udgt gt bc cmt. Cbb` W nbsts nbsts X Xs s 0: per oa md` prbvg`es 8 udgts bc prbtegds, p rbtegds, = udgt bc nmrhbfy`rmtes md` = udgt bc cmt. Cgd` but tfe iemst nbst nblhgdmtgbds nblhg dmtgbds bc tfe cbb`s leetgda tfe requgreledts.

].3

^uppbse bde arml bc wfemt prbvg`es :.= a bc prbtegds md` :.85 a bc nmrhbfy`rmtes md` tfe nbrrespbd`gda vmiues cbr rgne mre :.:5 a md` :.5 a respentgveiy. Qfemt nbsts Xs 5 md` rgne nbsts Xs 8: per oa. o a. Pfe P fe lgdgl ul `mgiy `mg iy requg reledt reledts s bc prb prbtegd tegds s md` nmrhbfy nmrhbfy`rmt `rmtes es cbr md mve rmae rma e lmd mre 5: 5 : a md` m d` 8 8:: :: a r espentgveiy. Gd wfmt wfmt qumdtgtges sfbui` wfemt md` md` rgne rgne he lgxe` gd tfe `mgiy `get tb prbvg`e tfe lgdglul `mgiy requgreledts bc prbtegds md` nmrhbf nmr hbf y`rmtes mt lgdglul nbst, m ssulgda tfmt hbtf wfemt md` rgne mre tb he tmoed gd tfe `get2 `get2

].1

M amr`ed amr`eder er fms fms m suppiy suppiy bc cer certgigze tgigzerr bc ty type pe G wfgnf nbdsgst s bc =:% dgtr baed md` 3% pfbspfbrgn mng`, md` type GG wfgnf nbdsgsts bc 5% dgtrbaed md` =:% pfbspfbrgn mng`. Mcter testgda tfe sbgi nbd`gtgbds, he cgd`s tfmt fe dee`s mt iemst =0 oa bc dgtrbaed md` =0 oa bc  pfbspf brgn mng` cbr fgs nrbp . Gc tfe t ype-G certgigzer n nbst bsts s 3: p pmgse mgse per per oa md` md` tfe tfe typ type-G e-GGG certgigzer nbsts 0: pmgse per oa, `eterlgde fbw lmdy ogibarmls ogiba rmls bc emnf certgigzer sfbui` he use` sb tfmt tfe dutrgedt requgreledts mre let mt m lgdglul lgdg lul nbst. Qfmt gs tfe lgdglul lg dglul nbst2

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IGDEMX ZXBAXMLLGDA

1

].6

M lmd bwds m cgei` bc mrem =::: sq l. Fe wmdts tb pimdt crugt trees gd gt. Fe fms m sul bc Xs =0:: tb purnfmse ybuda trees. Fe fms tfe nfbgne nfbgn e bc twb types bc trees. Pype Pype M requgres =: sq l bc arbud` per tree md` nbsts Xs 8: per tree md` type H requgres 8: sq. l bc arbud` per tree md` nbsts Xs 85 per tree. Qfed Q fed cuiiy arbwd, m tree bc typ type e M prb`unes md mvermae bc 8: oa bc crugt wfgnf nmd he sbi` mt m prbcgt prbcgt bc Xs 8 p er o a, md` t ype H prb`une prb `une s md mvermae mver mae bc  0: oa bc crugt wfgnf nmd he sbi` mt m prbcgt bc Xs =.5: per oa. Fbw lmdy bc emnf type sfbui` he pimdte` tb mnfgeve lmxglul prbcgt wfed tfe trees mre cuiiy a arbwd2 rbwd2 Qfmt gs tfe lmxglul prbcgt 2

].7

Md me rbp imde im de nm d nmrry nmr ry m l mxg lul b c 8:: pmssedaers. p mssedaers. M prbcgt b bcc Xs 0:: gs lm`e bd emnf cgrst nimss nimss tgnoet md` md` m prbcgt bc Xs Xs >:: gs lm`e lm`e bd emnf en bdbly bdbl y ni mss t gno et. P fe mg rig de reserves mt iemst 8: semts cbr cgrst nimss. Fbwever, mt iemst cbur tgles ms lmdy pm s se da e rs precer tb trmvei trmvei hy enbdbly nimss nimss tb tfe cgrst nimss. @eterlgde @eterlgde fbw fbw lmdy bc  emnf emnf type bc tg noets lust he s bi` gd br`er tb lmxglgze lmxglgze tfe tfe prbcgt cbr tfe mgrigde. Qfmt gs tfe lmxglul prbcgt 2

].=:

M cmntbry bwder fms tb purnfmse twb types bc lmnfgdes, M md` H, cbr fgs cmntbry. Pfe requgreledts md` iglgtmtgbds cbr tfe lmnfgdes mre ms cbiibws ;  Mrem bnnupge` hy Imhbur requgre` @mgiy butput gd tfe lmnfgde cbr emnf lmnfgde udgts Lmnfgde M

=::: sq l

=8 led

3:

Lmnfgde H

=8:: sq l

6 led

0:

Fe fms md mrem bc 7::: sq l mvmgimhie md` 18 sogiie` led l ed tb bpermte tfe lmnfgdes. Fbw lmdy lmnfgd lmn fgdes es bc emnf t ype sf bui` he hu huy y tb lmxg lgze tfe `mgiy butput butput 2 ].==

M lmducmnture prb`unes twb types bc steei trudos. Fe fms twb lmnfgdes, M md` H. Pfe cgrst type bc trudo requgres > fburs bd lmnfgde M md` > fburs bd lmnfgde H. Pfe senbd` type requgres > fburs bd lmnfgde M md` 8 fburs bd lmnfgde H. Lmnfgdes M md` H nmd wbro cbr mt lbst =6 fburs md` =5 fburs per `my respentgveiy. Fe emrds m prbcgt bc Xs =5 md` Xs 85 per trudo bd tfe cgrst type md` senbd` type respentgveiy. Fbw lmdy trudos bc emnf type lust fe lmoe emnf `my tb lmoe lmxglul prbcgt 2

].=8

M tby nblpmdy lmducmntures twb types bc `biis M md` H. Lmroet tests md` mvmgimhie resburnes fmve gd`gnmte` tfmt tfe nblhgde` prb`untgbd ievei sfbui` dbt d bt exnee` =8:: `biis per weeo md` tfe `elmd` cbr `biis `bi is bc type H gs mt lbs lbstt fmic bc tfmt cbr `biis bc type M. Curtfer, Curtfer, tfe prb`untgbd ievei bc `biis `bii s bc type M nmd exnee` > tgles tfe prb`untgbd bc `bii `biis s bc btfer type hy mt lbst 3:: udgts. Gc tfe nblpmdy lmoes prbcgt bc Xs =8 md` Xs =3 per. `bii respentgveiy bd `biis `bii s M md` H, fbw lmdy bc emnf sfbui` he prb`u prb`une` ne` wee weeoiy oiy gd br`er tb lmxglgze lmxglgze tfe prbcgt 2

].=>

M crugt arbwer nmd use twb types bc certgigzer certgig zer gd fgs amr`ed, hrm hrmd` d` Z md` hrmd` ]. Pfe mlbudts (gd oa) bc dgtrbaed, pfbspfbrgn mng`, pbtmsf md` nfibrgde gd m hma bc emnf hrmd` mre agved gd tfe tmhie. Pests gd`gnmte tfmt tfe amr`ed dee`s mt iemst 80: oa bc pfbspfbrgn mng`, mt iemst i emst 81: oa bc pbtmsf md` mt lbst >=: oa bc nfibrgde. Gc tfe a rbwer wmdts tb lg dgl gze t fe mlb udt bc dgtrbaed m``e` tb tfe amr`ed, amr`ed, f fbw bw lmdy hmas hma s bc emnf hrmd` sfbui` he use` 2 Qfmt g s tfe lgdglul mlbudt bc dgtr baed m``e` tb tfe amr`ed a mr`ed 2 Hrmd` Z ]

Oa per hma Dg Dgtr trba baed ed Zf Zfbs bspf pfbr brgn gn mng` mng` Zb Zbtm tmsf sf Nfib Nfibrg rgde de > = > =.5 >.5 8 =.5 8

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IGDEMX ZXBAXMLLGDA

6

].=0

M nbbpermtgve sbngety bc cmrlers fms 5: fentmres bc imd` tb arbw twb nrbps V md` W. Pfe prbcgt crbl nrbp nrbps s V md` W per f fent entmre mre mre estglmt estg lmte` e` ms Xs =:5:: md` Xs 7::: respentgveiy. Pb nbdtrbi nbdtrbi wee`s, m igqug` ferhgng`e fms tb he use` crb nrbps V md` W mt tfe rmte bc 8: igtres md` =: iigtres gtres per fentmre. Curtfer db lbre tfmd 6:: igtres bc ferhgng`e sfbui` he use` gd br`er tb prbtent cgsf md` wgi` igce usgda m pbd` wfgnf nbiients `rmgdmae crbl tfgs imd`. Fbw lunf imd` sfbui` sfbu i` he mi ibn mte` tb em nf nrb p sb ms tb lmxg l mxg lgze lgz e tfe tbtmi tbtmi prb prbcgt cgt bc tfe sbnget sbngety y2

].=5

M le`gnmi nblpmdy fms cmntbrges mt twb pimnes, M md` H. Crbl tfese pimnes suppiges mre lm`e tb emnf emnf bc bc gts tfree tfree maednges maednges sgtumte` sgtumte` mt Z, ] md` X. P fe lbdt fiy requgre requ gre ledt s b c  tfe ma e dn ge s mr e re sp spent ent g vei y 0: , 0: md` 5: pmn pmnoet oets s bc tfe le`gngde le`gngdes, s, wf wfgie gie tfe prb`un prb`untgb tgbd d nmpmngty bc tfe cmntbrges, M md` H, mre 3: md` 1: pmnoets respentgveiy. Pfe trmdspbrtmtgbd nbst p er pmnoet crbl tfe c mntbrge s tb tfe mae dnges mre agved agved heib heibw; w;    

Prmdspbrtmtgbd nbst per pmnoet (gd Xs) Crbl  M H Pb Z 5 0 ] 0 8 X > 5

Fbw lmdy pmnoets crbl emnf cmntbry sfbui` he trmdspbrte` tb emnf maedny sb tfmt tfe nbst bc trmdspbrtmtgbd gs lgdglul 2 Misb, cgd` tfe lgdglul nbst. ].=3

Md bgi bgi nbl nblpmd pmdy y fms ttwb wb `epbt `epbts, s, M md` md` H, wgtf nmpmngtg es bc 1::: ig tres md` 0::: igt res respentgveiy. Pfe nblpmdy fms tb suppiy bgi tb tfree petrbi pulps, @, E md` C, wfbse requgr eledts mre 05:: igtres, >::: igtres md` >5:: igtres respentgveiy. Pfe `gstmdne hetweed requgreledts tfe `epbts md` tfe petrbi pulps gs agved ag ved gd tfe cbiibwgda tmhie ;    

@gstmdne (gd ol) Crbl  M

Pb @ E C

1 3 5

H > 0 8

Mssulgda tfmt tfe trmdspbrtmtgbd nbst per ol gs Xs 8 per igtre, ig tre, fbw sfbui` tfe `eigvery he snfe`uie` gd br`er tfmt tfe trmdspbrtmtgbd nbst gs lgdglul 2 ].=1

M nblpmdy lmoes twb ogd`s bc iemtfer heits, M md` H. Pfe respentgveiy prbcgts mre Xs 0 md` Xs > per heit. Emnf heit bc type M requgres r equgres twgne ms lunf tgle ms m heit h eit bc type H. Gc mii tfe hei heits ts were bc type H, tfe nblpmdy nbui` lmoe =::: heits per `my. Pfe suppiy bc iemtfer gs succgngedt cbr bdiy 6:: heits per `my (hbtf types bc heits M md` H requgre tfe smle mlbudt bc  iemtfer). 0:: hunoies mre mvmgimhie cbr heits bc type M, md` 1:: hunoies mre mvmgimhie mvmgimh ie cbr type H per `my. Qfmt sfbui` he tfe `mgiy prb`untgbd bc emnf type bc heit sb ms tb lmxglgze tfe prbcgt 2

].=6

M tby lmducmnture prb`unes twb types bc `biis, M md` H. Emnf `bii bc type H tmoes twgne ms ibda tb prb`une ms bde `bii bc type M. Pfe nblpmdy fms tgle tb lmoe m lmxglul bc 8::: `biis bc type M per `my. Pfe suppiy bc pimstgn gs succgngedt tb prb`une =5:: `biis per `my md` emnf type requgres md equmi mlbudt mlb udt bc gt. Cbr type H, bdiy 3:: `resses mre mvmgimhie per `my. `my. Gc tfe nblpmdy lmoes m prbcgt bc Xs > md` Xs 5 per `bii respentgveiy bd `biis bc type M md` type H, fbw lmdy bc emnf sfbui` he prb`une` per `my gd br`er tb lmxglgze tfe prbcgt 2

].=7

Gc m ybuda lmd rg`es fgs lbtbrnynie mt 85 ol/f, fe fms tb sped` Xs 8 per ol bd petrbi. Gc fe rg`es gt mt m cmster spee` bc 0: ol/f, tfe petrbi nbst nbs t gdnremses tb Xs 5 per ol. Fe fms Xs =:: tb sped` bd petrbi md` wgsfes tb cgd` wfmt gs tfe lmxglul `gstmdne fe nmd trmvei wgtfgd bde fbur. Express gt ms m igdemr prbar prbarmllgda mllgda prbhiel md` sbiv s bive e gt armpfgnmi armpfg nmiiy. iy.

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IGDEMX ZXBAXMLLGDA

7

HBMX@ ZXBHIE^ EVEXNG^E „ GG

].=

].8

].>

].0

M prb`uner fms >: md` =1 udgts bc imhbur md` nmpgtmi respentgveiy wfgnf fe nmd use tb prb`une prb`un e twb type bc abb`s V md` W. W. P Pb b prb`une bde udgt udg t bc V, 8 udgts bc imhbur md` > udg udgts ts bc  nmpgtmi mre requgre`. ^glgimri ^g lgimriy y, > udgts bc imhbur md` = udgt udgt bc nmpgtmi ggs s requgre` tb prb`une bde udgt bc W. Gc V md` W mre prgne` mt Xs =:: md` =8: per udgt respentgveiy, fbw sfbui` tfe prb`uner use fgs resburnes tb lmxglgse tfe tbtmi revedue r evedue 2 ^bive tfe prbhiel armpfgnmiiy. _N.H.^.E. 8:::\ M nblpmdy lmducmntures twb types bc tbys M md` H. Pype M requgres 5 lgdutes emnf cbr nuttgda md` =: lgdutes emnf cbr msselhigda. msselhi gda. Pype Pype H requgres 6 lgdutes emnf cbr nuttgda md` 6 lgdutes emnf cbr msselhigda. Pfere ms > fburs mvmgimhie mvmgimhi e cbr nuttgda md` 0 fburs mvmgimhie mvmgimhi e cbr msselhigda gd m `my. Pfe prbcgt gs Xs 5: emnf bd type M md` Xs 3: emnf bd type H. Fbw lmdy tbys bc emnf type sfbui` tfe nblpmdy lmducmnture gd m `my tb lmxglgse tfe prbcgt 2 _N.H.^.E. 8::=\ M nblpmdy seiis twb `gccere ` gcceredt dt prb`unts M md` H. Pfe twb prb`unts mre prb`une` gd m nbllbd prb`untgbd prbness wfgnf fms m tbtmi nmpmngty nmpmng ty bc 5:: lmd fburs. Gt tmoes 5 fburs tb prb`une m udgt bc M md` > fburs tb prb`une m udgt bc H. P Pfe fe `elmd` gd tfe lmroet sfbws, tfmt tfe lmxglul dulher bc udgts bc M tfmt nmd he sbi` gs 1: md` tfmt cbr H gs =85. Zrbcgt bd emnf udgt bc M gs Xs 8: md` tfmt bd H gs Xs =5. Fbw lmdy udgts bc M md` H sfbui` he prb`une` tb mxglgse tfe prbcgt 2 ^bive gt g t armpfgnmiiy. _N.H.^.E. 8::>\ M lmducmnturer lmoes twb types bc nups, M md` H. Pfree lmnfgdes mre requgre` tb lmducmnture tfe nups md` tfe tgle gd lgdutes requgre` hy emnf gs ms agved heibw ;

Pype bc Nup  M H

].5

].3

].1

].6

Lmnfgdes G =8 3

GG =6 :

GGG 3 7

Emnf lmnfgde gs mvmgimhie mvmgi mhie cbr m lmxglul p pergb` ergb` bc 3 fburs per `my. Gc tfe prbcgt bd emnf nup M gs 15 pmgse, md` bd H gt gs 5: pmgse, sfbw tfmt =5 nups bc type M md` >: nups bc type H sfbui` he lmducmnture` per `my tb aet tfe lmxglul prbcgt. _N.H.^.E. 8::>\ Oeiibaa gs m dew neremi cbrle` bc m lgxture bc hrmd md` rgne tfmt nbdtmgds mtiemst 66 armls bc  prbtegd md` mtiemst >3 lgiigarmls lgii garmls bc grbd. Odbwgda Odbwgda tfmt hrmd nbdtmgds 6: armls bc prbtegd md` 0: lgiigarmls bc grbd per ogibarml md` tfmt rgne nbdtmgds =:: armls bc prbtegd md` >: lgiigarmls bc grbd per ogibarml, cgd` cg d` tfe lgdglul nbst bc prb`ungda tfe dew neremi gc hrmd nbsts Xs 0 per ogibarml ogiba rml md` rgne nbsts Xs Xs 0 per ogibarml. _N.H.^.E. 8::>\ M lmd fms Xs =5:: cbr purnfmse bc rgne md` wfemt. M hma bc rgne md` m hma bc wfemt nbst Xs =6: md` Xs =8: respentgveiy respentgvei y. Fe fms m stbrmae nmpmngty bc =: hmas bdiy. Fe emrds m prbcgt bc  Xs == md` Xs 7 respentgveiy per hma bc rgne md` wfemt. Cbrluimte gt ms m igdemr prbarmllgda prbarmllg da prbhiel md` sbive gt armpfgnmiiy cbr lmxglul prbcgt. p rbcgt. _N.H.^.E. 8::0\ Mdgi wmdts tb gdvest mtlbst Xs =8::: gd Hbd`s M md` H. H. Mnnbr`gda Mnnbr`g da tb ruies, fe fms tb gdvest mti mtiemst emst Xs X s 8::: gd Hbd` M md` mtiemst Xs 0::: gd Hbd` H. Gc tfe rmte bc gdterest bd Hbd` M gs 6% p.m. md` bd Hbd` H gs =:% p.m. p.m.,, fbw sfbui` fe gdvest fgs lbdey cbr lmxglul lmxg lul gdterest 2 ^bive tfe prbhiel armpfgnmiiy armpfgnmii y. _N.H.^.E. 8::0\ M cgrl `emis wgtf tfe twb ogd`s bc crugt kugnes„pgde mppie md` brmdae kugne. Pfese mre lgxe` md` twb lgxtures mre sbi` sbi ` mre sbct `rgdos M md` H. Bde tgd bc M requgres requgr es 0 igtres bc pgde mppie mppi e md` = igtre bc brmdae kugne. Bde tgd bc H requgres 8 igtres bc pgde mppie md` > igtres bc brmdae  kugne. Pfe cgrl fms bdiy 03 03 igtres bc pgde mppie kugne md` md` 80 igtres bc brmdae kugne. Emnf Emnf tgd bc  M md` H mre sbi` mt m prbcgt bc Xs 0 md` Xs > respentgveiy. Fbw lmdy tgds bc emnf type sfbui` tfe cgrl prb`une tb lmxglgse tfe prbcgt 2 ^bive tfe prbhiel armpfgnmiiy armpfgnmiiy.. _N.H.^.E. 8::0\

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IGDEMX ZXBAXMLLGDA

].7

M lmducmnturer lmducmnturer lmoes twb prb`unts, M md` H. Zrb`unt M seiis mt Xs 8:: per udgt md` tmoes >: lgdutes tb lmoe. Zrb`unt H seiis mt Xs >:: per udgt md` tmoes = fbur tb lmoe. Pfere gs m perlmdedt br`er bc =0 udgts bc prb`unt M md` =3 udgts bc prb`unt H. M wbrogda weeo nbdsgsts bc 0: fburs bc prb`untgbd md` tfe weeoiy turd bver lust dbt he iess tfmd Xs =::::. Gc tfe prbcgt bd emnf bc prb`unt M gs Xs 8: md` bd prb`unt H gs Xs >:, tfed fbw lmdy bc emnf sfbui` he prb`une` sb tfmt tfe prbcgt gs lmxglul 2 Misb, cgd` tfe lmxglul prbcgt. ^bive tfe prbhiel armpfgnmiiy. _N.H.^.E. 8::5\

].=:

Pwb tmgibrs M md` H emrd Xs =5 = 5 md` Xs 8: per `my respentgveiy. respentg veiy. M nmd stgtnf 3 sfgrts md` 0 pmdts per `my, wfgie H nmd stgtnf =: sfgrts sfg rts md` 0 pmdts per `my. Fbw lmdy `mys sfmii emnf wbro gc gt gs `esgre` tb prb`une mtiemst 3: sfgrts md` >8 pmdts mt m lgdglul imhbur nbst 2 ^bive tfe prbhiel armpfgnmiiy. _N.H.^.E. 8::5\

].==

M cmrler fms m suppiy bc nfelgnmi certgigzers bc type M wfgnf nbdtmgds =:% dgtrbaed md` 3% pfbspfbrgn mng` md` bc type H wfgnf nbdtmgds 5% dgtrbaed dg trbaed md` =:% bc pfbspfbrgn pfbspfbrgn mng`. Mcter sbgi testgda gt g t gs cbud` tfmt mtiemst 1 oa bc dgtrbaed md` tfe smle q qumdtgty umdtgty bc pfbspfbrgn mng` gs requgre` cbr m abb` nrbp. Pfe certgigzer bc b c type M nbsts Xs 5 per oa md` tfe type H nbsts Xs 6 per oa. Usgda igdemr prbarmllgda cgd` fbw lmdy ogibarmls bc emnf type bc tfe certgigzer sfbui` he hbuaft tb leet tfe requgreledt md` tfe nbst he lgdglul. ^bive tfe prbhiel armpfgnmiiy. _N.H.^.E. 8::3\

].=8

M curdgture `emier `emis bdiy gd twb gtels „ tmhies md` nfmgrs. Fe fms Xs =:::: =:::: tb gdvest md` m spmne tb stbre mtlbst 3: pgenes. M tmhie nbsts fgl Xs 5:: md` m nfmgr Xs 8::. Fe nmd seii m tmhie mt m prbcgt prbcg t bc Xs 5: md` m nfmgr mt m prbcgt bc Xs =5. Mssule tfmt fe nmd seii sei i mii gtels g tels tfmt fe huys. Usgda igdemr igd emr prbarmllgda cbrluimte tfe prbhiel cbr lmxglul prbcgt md` sbive gt armpfgnmiiy. _N.H.^.E. 8::3\

].=>

M `emier wgsfes tb purnfmse m dulher bc cmds md` sewgda lmnfgdes. Fe fms bdi bdiy y Xs 513: tb gdvest md` fms spmne cbr mtlbst 8: gtels. M cmd md` sewgda lmnfgde nbst Xs >3: md` Xs 80: respentgveiy. Fe nmd seii m cmd mt m prbcgt p rbcgt bc Xs 88 md` sewg sewgda da lmnfgde mt m prbcgt bc Xs =6. Mssulgda tfmt fe nmd seii wfmtever fe huys, fbw sfbui` fe gdvest fgs lbdey gd br`er tb lmxglgse fgs f gs prbcgt 2 Prmdsimte tfe prbhiel gdtb I.Z.Z I.Z.Z md` sbive ggtt armpfgnmiiy._N.H.^.E. armpfgnmiiy._N.H.^.E. 8::3\

].=0

Gc m ybuda lmd rg`es fgs lbtbrnynie mt 85 ol/f, fe fm` tb sped` Xs 8 per ol bd petrbi. Gc fe rg`es mt m cmster spee` sp ee` bc 0: ol/f, tfe petrbi nbst gdnremse mt Xs 5 per ol. Fe fms Xs =:: tb sped` bd petrbi md` wgsfes tb cgd` cg d` wfmt gs tfe lmxglul `gstmdne fe nmd trmvei wgtfgd bde fbur.. Express tfgs ms bd I.Z fbur I.Z.Z .Z md` sbive gt armpfgnmiiy. _N.H.^.E. 8::1\

].=5

Md merbpimde nmd nmrry m lmxglul bc b c 8:: pmssedaers. M prbcgt bc Xs 0:: gs lm`e bd emnf cgrst cg rst nimss tgnoet md` m prbcgt bc Xs >:: gs lm`e bd emnf senbd` nimss tgnoet. Pfe mgrigde reserves mtiemst 8: semts cbr cgrst-nimss. Fbwever, Fbwever, tgnoets bc emnf type lust he sbi` tb lmxglgse prbcgt cbr tfe mgrigde. Cbrl md IZZ md` sbive gt armpfgnmiiy. armpfgnmiiy. _N.H.^.E. 8::6\

].=3

M cmrler fms m suppiy bc nfelgnmi certgigzer bc type M wfgnf nbdtmgds =:% dgtrbaed md` 5% pfbspfbrgn mng`, md` type H wfgnf nbdtmgds 3% dgtrbaed md` =:% pfbspfbrgn mng`. Mcter testgda tfe sbgi nbd`gtgbds nbd`g tgbds bc tfe cgei`, gt wms cbud` tfmt mt iemst =0 oa bc dgtrbaed md` =0 oa bc pfbspfbrgn mng` gs requgre` cbr prb`ungda m abb` nrbp. Pfe certgigzer certgi gzer bc type bc M nbsts Xs 5 per oa md` tfe type H nbsts Xs > per oa. Fbw lmdy oa bc emnf type bc tfe certgigzer sfbui` he use` tb leet tfe requgreledt mt tfe lgdglul pbssghie nbst 2 Usgda IZZ, sbive tfe mhbve prbhiel armpfgnmiiy armpfgnmii y. _N.H.^.E. 8::6\

].=1

M `get cbr m sgno sg no persbd lust nbdtmgd mt iemst 0::: udgts bc vg vgtmlgds, tmlgds, 5: udgts bc lgdermis md` =0:: udgts bc nmibrges. Pwb cbb`s M md` H mre mvmgimhie mvmgimhi e mt m nbst bc Xs 5 md` Xs 0 per udgt respentgveiy. Bde udgt bc tfe cbb` M nbdtmgds nbdtmgds 8:: udgts bc vgtmlg vgtmlgds, ds, = udgt bc lgdermis md` 0: udgts bc nmibrges, wfgie bde udgt bc tfe cbb` H nbdtmgds =:: udgts bc vgtmlgds, 8 udgts bc  lgdermis lgderm is md` md` 0: udgts bc nm nmibrges. ibrges. Cgd` wfmt wfmt nblhgd nbl hgd mtgbd mtg bd bc tfe cbb` cbb`s s M md` md ` H sfbui s fbui ` he use` tb fmve i emst n bst hut gt l ust smtgscy tfe re qugreledts bc tfe tfe sgno pers persbd. bd. Cbr Cbrl l tfe tfe questgbd ms IZZ md` sbive gt g t armpfgnmiiy. _N.H.^.E. 8::6\

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].=6

Bde ogd` bc nmoe requgres 8:: a bc cibur md` 85 a bc cmt md` mdbtfer ogd` bc nmoe requgres =:: a bc cibur md` 5: a bc cmt. Cgd` tfe lmxglul dulher bc nmoes wfgnf nmd he lm`e crbl 5 oa bc  cibur md` = oa bc cmt mssulgda tfmt tfere gs db sfbrtmae bc tfe btfer gdare`gedts use` gd lmogda tfe nmoes. Cbrluimte Cbrluimt e tfe mhbve ms m igdemr prbarmllgda prba rmllgda prbhiel md` sbive armpfgnmiiy. armpfgn miiy. _N.H.^.E. 8::7\

].=7

M `get gs tb nbdtmgd 6: udgts bc vgtmlgd M md` =:: udgts bc lgdermis. Pwb cbb`s C = md` C8 mre mvmgimhie. Cbb` C= nbsts Xs 0 per udgt md` C8 nbsts Xs 3 per udgt. Bde udgt bc cbb` C = nbdtmgds > udgts bc vgtmlgd M md` 0 udgts bc lgdermis. Bde udgt udg t bc cbb` C8 nbdtmgds 3 udgts bc vgtmlgd M md` > udgts bc lgdermis. Cbrluimte tfgs ms m igdemr prbarmllgda prbhiel md` cgd` armpfgnmiiy tfe lgdglu lg dglul l nbst cbr `g `get et tfmt nbdsgsts n bdsgsts bc lgxture bc tfese twb cbb`s md` md` misb leet tfe lgdglmi dutrgtgbdmi requgreledts.

].8:

_N.H.^.E. 8::7\

M slmii cgrl lmducmntures abi` rgdas md` nfmgds. Pfe tbtmi dulher bc rgdas md` nfmgds lmducmnture` lmducmntu re` per `my 1 gs mtlbst 80. Gt tmoes = fbur tb lmoe m rgda md` >: lgdutes tb lmoe m nfmgd. Pfe lmxglul dulher fburs mvmgimhie per `my gs =3. Gc tfe prbcgt bd m rgda gs Xs >:: md` tfmt bd m nfmgd gs Xs =7:, cgd` tfe dulher bc rgdas md` nfmgds tfmt sfbui` h he e lmducmnture` per `my, sb ms tfe emrd tfe lmxglul lmxg lul prbcgt. Lmoe gt g t ms md I.Z.Z I.Z.Z.. md` sbive gt armpfgnmi armpfgnmiiy. iy. _N.H.^.E. 8:=:\

].8=

M lernfmdt pimds tb seii twb types bc persbdmi nblputers „ m `esotbp lb`ei md` m prbtmhie lb`ei tfmt wgii nbst Xs. 85,::: md` Xs. 0:,::: respentgveiy. Fe estglmtes tfmt tfe tbtmi lbdtfiy `elmd` bc nblputers wgii dbt exnee` 85: udgts. @eterlgde tfe dulher bc udgts bc  emnf type bc nblputers wfgnf tfe lernfmdt sfbui` stbno tb aet lmxglul prbcgt gc g c tfe `bes dbt wmdt tb gdvest lbre tfmd Xs. 1: imofs md` fgs prbcgt bd tfe `esotbp lb`ei gs Xs. 0,5:: md` bd tfe pbrtmhie lb`ei gs Xs. 5,:::. Lmoe md I.Z.Z I.Z.Z.. md` sbi sbive ve gt armpfgnmiiy. _N.H.^.E. 8:==\

].88

M `getgngmd wgsfes wg sfes tb lgx twb types bc cbb`s gd sunf m wmy tfmt tfe vg vgtmlgd tmlgd nbdtedts bc tfe lgxture nbdtmgds mt iemst 6 udgts bc vgtmlgd M md` =: udgts bc vgtmlgd N. cbb` G nbdtmgds 8 udgts/oa bc vgtmlgd M md` = udgt/oa bc vgtmlgd N wfgie Cbb` GG nbdtmgds = udgts/oa bc vgtmlgd M md` 8 udgts /oa bc vgtmlgd N. Gt nbsts Xs. 5 per oa tb purnfmse purnfmse Cbb` G md` Xs. 1 per oa tb purnfmse Cbb` GG. @eterlgde tf tfe e lgdglul nbst bc sunf m lgxture. Cbrluimte tfe mhbve ms m IZZ md` sbive gt armpfgnmiiy.

].8>

_N.H.^.E. 8:=8\

M lmducmnturer nbdsg`ers tfmt led md` wbled wbroers mre equmiiy eccgngedt md` sb fe pmys tfel mt tfe smle rmte. Fe fms >: md` =1 udgts bc wbroers (lmie md` celmie) md` nmpgtmi respentgveiy, wfgnf fe uses tb prb`une twb types ty pes bc abb`s M md` H. Pb prb`une bde udgt bc  M, 8 wbroers md` > udgts bc nmpgtmi mre requgre` wfgie > wbroers md` = udgt bc nmpgtmi gs requgre` requgre ` tb prb`une bde udgt bc H. Gc M md` H mre prgne` mt Xs. =:: md` Xs. =8: per udgt respentgveiy, fbw sfbui` fe use fgs resburnes tb lmxglgse tfe tbtmi revedue 2 Cbrl tfe mhbve ms md IZZ md` sbive armpfgnmiiy. @b ybu maree wgtf tfgs tfg s vgew bc tfe lmducmnturer tfmt led md` wbled wbroers mre equmiiy eccgngedt md` sb sfbui` he pmg` mt tfe smle rmte 2

www.tfgdogg fgdoggt.gd

_N.H.^.E. 8:=>\

 

IGDEMX ZXBAXMLLGDA

=8

MD^QEX OEW

EVEXNG^E „ = (UD^BISE@ ZXBHIEL^) =5 51:  md` y < 0 , wfed x < 1 1

=.

Pfe lmxglul vmiue vmi ue bc z <

8.

=: rgdas md` 8: nfmgds ? lmxglul lmxg lul prbcgt < Xs 55::

>.

Abb` V < > udgts, abb` W < 6 udgts ? lmxglul revedue < Xs =83:

0. 5.

Zrb`unt M < 3: udgts, prb`unt H < 80: udgts ? lmxglul prbcgt < Xs =:8: Cbb` V < 3 udgts, Cbb` W < 3 udgts ? lgdglul nbst < Xs >::

3.

Db lgdglul vmiue

1.

Certgigzer type = < =:: oa, certgigzer type GG < 6: oa ? lgdglul nbst < Xs 78

6.

Pype M < 8: trees, typ e H < 0: trees ? lmxglul prbcgt < Xs >8::

7.

Cgrst nimss < 0: tgnoets, enbdbly nimss < =3: tgnoets ? lmxglul prbcgt < Xs 30,:::

=:.. =:

Pype M < 3 lmnfgdes, type H < : lmnfgdes ? lmxglul lmxg lul butput < >3: udgts

==.. ==

Pype G < : trudos, type GG < 3 trudos? lmxglul prbcgt < Xs =5:.

=8.. =8

Pype M .. =>

Hrmd` Z < 0: hmas, hrmd` ] < =:: hmas ? lgdglul mlbudt bc dgtrbaed < 01: oa

=0.. =0

Nrbp V < >: fentmres, Nrbp W < 8: fentmre ? lmxglul lmxg lul prbcgt < Xs. 0,75,:::

=5.. =5

Lgdglul nbst < Xs 0:: Db. bc hmnoets trmdspbrte`  Maedny Z Maedny ] Maedny X Cmntbry M =: : 5: Cmntbry H >: 0: :

EVEXNG^E „ 8 (HBMX@ ZXBHIEL^) =. =83:, (>, 6)

8. =5::, (=8, =5)

>. M ; 85 udgts, H ; =85 udgts

5. Xs 0.3, (.3,.0)

3. =::, (5, 5)

1. (8:::, =::::)

6. M ; 7 tgd, H ; 5 tgd

7. =00:, (06, =3)

==. 51:, (5:, 0:)

=8. =:::, (8:, :)

=:. M ; 5 `mys, H ; > `mys

=>. >78, (6, =8)

ã 5: 0: óñ =0. >:, á , æ  > >  ö

=3. M ; 6: oa, H ; =:: Oa

=1. M ; 5 udgt, H ; >: udgt

=6. >:

8=. ==5::::, (8::, 5:)

88. >6, (8, 0)

=7. =:0

www.tfgdogg fgdoggt.gd

8:. 500:, (6, =3)